- Cauchy sequence
mathematics, a Cauchy sequence, named after Augustin Cauchy, is a sequencewhose elements become "arbitrarily close to each other" as the sequence progresses. To be more precise, by dropping enough (but still only a finite number of) terms from the start of the sequence, it is possible to make the maximum of the distances from any of the remaining elements to any other such element smaller than any preassigned positive value.
In other words, suppose a pre-assigned positive real value is chosen.However small is, starting from a Cauchy sequence and eliminating terms one by one from the start, after a finite number of steps, any pair chosen from the remaining terms will be within distance of each other.
The utility of Cauchy sequences lies in the fact that in a
complete metric space(one where all such sequences are known to converge to a limit), they give a criterion for convergence which depends only on the terms of the sequence itself. This is often exploited in algorithms, both theoretical and applied, where an iterative process can be shown relatively easily to produce a Cauchy sequence, consisting of the iterates.
The notions above are not as unfamiliar as might at first appear. The customary acceptance of the fact that any real number "x" has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers (whose terms are the successive truncations of the decimal expansion of "x") has the real limit "x". In some cases it may be difficult to describe "x" independently of such a limiting process involving rational numbers.
Generalizations of Cauchy sequences in more abstract
uniform spacesexist in the form of Cauchy filterand Cauchy net.
Cauchy sequence of real numbers
of real numbers is called Cauchy, if for every positive real number "ε", there is a positive
integer"N" such that for all natural numbers "m","n" > "N"
where the vertical bars denote the
In a similar way one can define Cauchy sequences of complex numbers.
Cauchy sequence in a metric space
To define Cauchy sequences in any metric space, the absolute value is replaced by the "distance" between and .
Formally, given a
metric space("M", "d"), a sequence
is Cauchy, if for every positive
real number"ε" > 0 there is a positive integer"N" such that for all natural numbers "m","n" > "N", the distance
is less than "ε". Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in "M". Nonetheless, such a limit does not always exist within "M".
A metric space "X" in which every Cauchy sequence has a limit (in "X") is called complete.
real numbers are complete, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers.
A rather different type of example is afforded by a metric space "X" which has the discrete metric ( where any two distinct points are at distance "1" from each other). Any Cauchy sequence of elements of "X" must be constant beyond some fixed point, and converges to the eventually repeating term.
Counter-example: rational numbers
rational numbers Q are not complete (for the usual distance):There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having no limit in Q. In fact,if a real number "x" is irrational, then the sequence ("x""n"), whose "n"-th term is the truncation to "n"decimal places of the decimal expansion of "x", gives Cauchy sequence of rational numbers with irrational limit "x". Irrational numbers certainly exist, for example:
* The sequence defined by "x0 = 1", "xn+1" = ("xn" + 2/"xn")/2 consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; however it converges to the irrational square root of two, see Babylonian method of computing square root.
* The sequence of ratios of consecutive
Fibonacci numbers which, if it converges at all, converges to a limit satisfying , and no rational number has this property. If one considers this as a sequence of real numbers, however, it converges to the real number , the Golden ratio, which is irrational.
* The values of the exponential, sine and cosine functions, exp("x"), sin("x"), cos("x"), are known to be irrational for any rational value of "x"≠0, but each can be defined as the limit of a rational Cauchy sequence, using, for instance, the
* Every convergent sequence (with limit "s", say) is a Cauchy sequence, since, given any real number "r" > 0, beyond some fixed point, every term of sequence is within distance "r"/2 of "s", so any two terms of the sequence are within distance "r" of each other.
* Every Cauchy sequence of real (or complex) numbers is bounded ( since for some "N", all terms of the sequence from the "N"-th onwards are within distance "1" of each other, and if "M" is the largest absolute value of the terms up to and including the "N"-th, then no term of the sequence has absolute value greater than "M"+"1").
* In any metric space, a Cauchy sequence which has a convergent subsequence with limit "s" is itself convergent (with the same limit), since, given any real number "r" > 0, beyond some fixed point in the original sequence, every term of the subsequence is within distance "r"/"2" of "s", and any two terms of the original sequence are within distance "r"/"2" of each other, so every term of the original sequence is within distance "r" of "s".
These last two properties, together with a lemma used in the proof of the
Bolzano-Weierstrass theorem, yield one standard proof of the completeness of the real numbers, closely related to both the Bolzano-Weierstrass theorem and the Heine–Borel theorem. The lemma in question states that every bounded sequence of real numbers has a convergent subsequence. Given this fact, every Cauchy sequence of real numbers is bounded, hence has a convergent subsequence, hence is itself convergent. It should be noted, though, that this proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of "constructing" the real numbers as the ml|Completion_%28metric_space%29|Completion|completion of the rational numbers, makes the completeness of the real numbers tautological.
One of the standard illustrations of the advantage of being able to work with Cauchy sequences and make use of completeness is provided by consideration of the summation of an
infinite seriesof real numbers(or, more generally, of elements of any complete normed linear space, or Banach space). Such a series is considered to be convergent if and only if the sequence of partial sums is convergent, where . It is a routine matter to determine whether the sequence of partial sums is Cauchy or not,since for positive integers "p" > "q" , :::::::.
If is a
uniformly continuousmap between the metric spaces "M" and "N" and ("x""n") is a Cauchy sequence in "M", then is a Cauchy sequence in "N". If and are two Cauchy sequences in the rational, real or complex numbers, then the sum and the product are also Cauchy sequences.
Cauchy sequences in topological vector spaces
There is also a concept of Cauchy sequence for a
topological vector space: Pick a local basefor about 0; then () is a Cauchy sequence if for all members of , there is some number such that whenever is an element of . If the topology of is compatible with a translation-invariant metric, the two definitions agree.
Cauchy sequences in topological groups
Since the topological vector space definition of Cauchy sequence only requires that there is a continuous "subtraction" operation, it can just as well be stated in the context of a
topological group: A sequence in a topological group is a Cauchy sequence if there for every open neighbourhood of the identity in exists some number such that whenever it follows that . As above, it is sufficient to check this for the neighbourhoods in any local base of the identity in .
As in the construction of the completion of a metric space, one can furthermore define the binary relation on Cauchy sequences in that and are "equivalent" if there for every open neighbourhood of the identity in exists some number such that whenever it follows that . This relation is an
equivalence relation. More precisely, it is reflexive since the sequences are Cauchy sequences. It is symmetric since which by continuity of the inverse is another open neighbourhood of the identity. It is transitive since where and are open neighbourhoods of the identity such that ; such pairs exist by the continuity of the group operation.
Cauchy sequences in groups
There is also a concept of Cauchy sequence in a group :Let be a decreasing sequence of normal subgroups of of finite index.Then a sequence in is said to be Cauchy (w.r.t. )
if and only iffor any there is such that .
Technically, this is the same thing as a topological group Cauchy sequence for a particular choice of topology on , namely that for which is a local base.
The set of such Cauchy sequences forms a group (for the componentwise product), and the set of null sequences (s.th. ) is a normal subgroup of . The
factor groupis called the completion of with respect to .
One can then show that this completion is isomorphic to the
inverse limitof the sequence .
An example of this construction, familiar in
number theoryand algebraic geometryis the construction of the "p-adic completion" of the integers with respect to a prime "p." In this case, "G" is the integers under addition, and "H""r" is the additive subgroup consisting of integer multiples of "p""r".
If is a cofinal sequence (i.e., any normal subgroup of finite index contains some ), then this completion is
canonicalin the sense that it is isomorphic to the inverse limit of , where varies over "all" normal subgroups of finite index.For further details, see ch. I.10 in Lang's "Algebra".
In constructive mathematics
constructive mathematics, Cauchy sequences often must be given with a "modulus of Cauchy convergence" to be useful. If is a Cauchy sequence in the set , then a modulus of Cauchy convergence for the sequence is a function from the set of natural numbers to itself, such that .
Clearly, any sequence with a modulus of Cauchy convergence is a Cauchy sequence. The converse (that every Cauchy sequence has a modulus) follows from the
well-ordering propertyof the natural numbers (let be the smallest possible in the definition of Cauchy sequence, taking to be ). However, this well-ordering property does not hold in constructive mathematics (it is equivalent to the principle of excluded middle). On the other hand, this converse also follows (directly) from the principle of dependent choice(in fact, it will follow from the weaker AC00), which is generally accepted by constructive mathematicians. Thus, moduli of Cauchy convergence are needed directly only by constructive mathematicians who (like Fred Richman) do not wish to use any form of choice.
That said, using a modulus of Cauchy convergence can simplify both definitions and theorems in constructive analysis. Perhaps even more useful are "regular Cauchy sequences", sequences with a given modulus of Cauchy convergence (usually or ). Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (in the sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved without using any form of the axiom of choice. Regular Cauchy sequences were used by
Errett Bishopin his [http://books.google.com/books?id=Z7I-AAAAIAAJ&dq=intitle:Foundations+intitle:of+intitle:constructive+intitle:analysis&lr=&as_brr=0&pgis=1 Foundations of Constructive Analysis] , but they have also been used by Douglas Bridgesin a non-constructive textbook (ISBN 978-0-387-98239-7). However, Bridges also works on mathematical constructivism; the concept has not spread far outside of that milieu.
Modes of convergence (annotated index)
*cite book | last = Spivak | first = Michael | authorlink = Michael Spivak | title = Calculus | year = 1994 | edition = 3rd ed. | location = Berkeley, CA | publisher = Publish or Perish | id = ISBN 0-914098-89-6 | url = http://www.mathpop.com/bookhtms/cal.htm
* (for uses in constructive mathematics)
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